Given that ABC is a right triangle.
The measure of ∠A is 45° and AB = 9
We need to determine the equations that could be used to solve the unknown lengths of ΔABC
Option A: [tex]\sin \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
The length of BC can be determined using the trigonometric ratios.
[tex]sin\ \theta=\frac{opp}{hyp}[/tex]
where [tex]\theta=45^{\circ}[/tex], [tex]opp= BC[/tex] and [tex]hyp = 9[/tex]
Hence, substituting the values, we get;
[tex]\sin \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
Hence, Option A is the correct answer.
Option B: [tex]\sin \left(45^{\circ}\right)=\frac{9}{BC}[/tex]
The length of BC can be determined using the trigonometric ratios.
[tex]sin\ \theta=\frac{opp}{hyp}[/tex]
where [tex]\theta=45^{\circ}[/tex], [tex]opp= BC[/tex] and [tex]hyp = 9[/tex]
Hence, substituting the values, we get;
[tex]\sin \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
Thus, the length of BC can be determined using [tex]\sin \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
Hence, Option B is not the correct answer.
Option C: [tex]9 \tan \left(45^{\circ}\right)=A C[/tex]
The length of AC can be determined using the trigonometric ratios.
[tex]tan \ \theta= \frac{opp}{adj}[/tex]
where [tex]\theta=45^{\circ}[/tex], [tex]opp= BC[/tex] and [tex]adj=AC[/tex]
Substituting the values, we get;
[tex]tan \ 45^{\circ}=\frac{BC}{AC}[/tex]
Thus, the length of AC using the trigonometric ratios is [tex]tan \ 45^{\circ}=\frac{BC}{AC}[/tex]
Hence, Option C is not the correct answer.
Option D: [tex](A C) \sin \left(45^{\circ}\right)=B C[/tex]
The formula for [tex]sin \ \theta[/tex] is given by the formula,
[tex]sin\ \theta=\frac{opp}{hyp}[/tex]
where [tex]\theta=45^{\circ}[/tex], [tex]opp= BC[/tex] and [tex]hyp = 9[/tex]
Hence, substituting the values, we get;
[tex]\sin \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
Thus, the given equation [tex](A C) \sin \left(45^{\circ}\right)=B C[/tex] is wrong.
Hence, Option D is not the correct answer.
Option E: [tex]\cos \left(45^{\circ}\right)=\frac{BC}{9}[/tex]
The formula for [tex]cos \ \theta[/tex] is given by the formula,
[tex]cos \ \theta=\frac{adj}{hyp}[/tex]
where [tex]\theta=45^{\circ}[/tex], [tex]adj=AC[/tex] and [tex]hyp = 9[/tex]
Substituting the values, we get;
[tex]\cos \left(45^{\circ}\right)=\frac{AC}{9}[/tex]
Hence, the given equation [tex]\cos \left(45^{\circ}\right)=\frac{BC}{9}[/tex] is not possible.
Thus, Option E is not the correct answer.
